Nuprl Lemma : cubical-path-app-1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[t:{X ⊢ _:(Path_A a b)}]. (t @ 1(𝕀) = b ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
cubical-path-app: pth @ r,
path-type: (Path_A a b),
interval-1: 1(𝕀),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
interval-1: 1(𝕀),
cubical-path-app: pth @ r,
cubicalpath-app: pth @ r,
cubical-app: app(w; u),
path-type: (Path_A a b),
cubical-term: {X ⊢ _:A},
implies: P ⇒ Q,
cubical-term-at: u(a),
cubical-type-at: A(a),
cubical-subset: cubical-subset,
pi1: fst(t),
and: P ∧ Q
Lemmas referenced :
cubical-term-equal2,
cubical-path-app_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
interval-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
path-type_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
independent_isectElimination,
lambdaFormation_alt,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
setElimination,
rename,
productElimination,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[t:\{X \mvdash{} \_:(Path\_A a b)\}]. (t @ 1(\mBbbI{}) = b)
Date html generated:
2020_05_20-PM-03_17_04
Last ObjectModification:
2020_04_06-PM-06_32_27
Theory : cubical!type!theory
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